Fick's laws of diffusion

Molecular diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: A single molecule moves around randomly. Middle: With more molecules, there is a clear trend where the solute fills the container more and more uniformly. Bottom: With an enormous number of solute molecules, randomness becomes undetectable: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. This smooth flow is described by Fick's laws.

Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low concentration across a concentration gradient. In one (spatial) dimension, the law is:

J=−Ddφdx{\displaystyle J=-D{\frac {d\varphi }{dx}}}

where

J is the "diffusion flux," of which the dimension is amount of substance per unit area per unit time, so it is expressed in such units as mol m−2 s−1. J measures the amount of substance that will flow through a unit area during a unit time interval.

D is the diffusion coefficient or diffusivity. Its dimension is area per unit time, so typical units for expressing it would be m2/s.

φ (for ideal mixtures) is the concentration, of which the dimension is amount of substance per unit volume. It might be expressed in units of mol/m3.

x is position, the dimension of which is length. It might thus be expressed in the unit m.

D is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6 × 10−9 to 2 × 10−9 m2/s. For biological molecules the diffusion coefficients normally range from 10−11 to 10−10 m2/s.

In two or more dimensions we must use ∇, the del or gradient operator, which generalises the first derivative, obtaining

J=−D∇φ{\displaystyle \mathbf {J} =-D\nabla \varphi }

where J denotes the diffusion flux vector.

The driving force for the one-dimensional diffusion is the quantity −∂φ/∂x, which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law (one-dimensional case) can be written as:

where erfc is the complementary error function. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space (i. e., corrosion product layer) is semi-infinite – starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is infinite (lasting both through the layer with n(x,0) = 0, x > 0 and that with n(x,0) = n0, x ≤ 0), then the solution is amended only with coefficient 1⁄2 in front of n0 (this might seem obvious, as the diffusion now occurs in both directions). This case is valid when some solution with concentration n0 is put in contact with a layer of pure solvent. (Bokstein, 2005) The length 2√Dt is called the diffusion length and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t (Bird, 1976).

As a quick approximation of the error function, the first 2 terms of the Taylor series can be used:

Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, poroussoils, population dynamics, nuclear materials, semiconductor doping process, etc. Theory of all voltammetric methods is based on solutions of Fick's equation. A large amount of experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition. In the vicinity of glass transition the flow behavior becomes "non-Fickian". It can be shown that the Fick's law can be obtained from the Maxwell-Stefan equations[1] of multi-componentmass transfer. The Fick's law is limiting case of the Maxwell-Stefan equations, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell-Stefan equations are used. See also non-diagonal coupled transport processes (Onsager relationship).

P is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.

c2 − c1 is the difference in concentration of the gas across the membrane for the direction of flow (from c1 to c2).

Fick's first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.

When two miscible liquids are brought into contact, and diffusion takes place, the macroscopic (or average) concentration evolves following Fick's law. On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and molecular scale, where molecular random walks take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.[3]

In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a tautology, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by renormalizing the fluctuating hydrodynamics equations.

In one dimension, the following derivation is based on a similar argument made in Berg 1977 (see references).

Consider a collection of particles performing a random walk in one dimension with length scale Δx and time scale Δt. Let N(x,t) be the number of particles at position x at time t.

At a given time step, half of the particles would move left and half would move right. Since half of the particles at point x move right and half of the particles at point x + Δx move left, the net movement to the right is:

An important example is the case where φ is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant D, the solution for the concentration will be a linear change of concentrations along x. In two or more dimensions we obtain

In 1855, physiologist Adolf Fick first reported[4] his now-well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of Thomas Graham, which fell short of proposing the fundamental laws for which Fick would become famous. The Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists: Darcy's law (hydraulic flow), Ohm's law (charge transport), and Fourier's Law (heat transport).

Fick's experiments (modeled on Graham's) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Fick's work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally possible.[5] Today, Fick's Laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does not follow Fick's laws (which does happen),[6][7] it is referred to as non-Fickian, in that they are exceptions that "prove" the importance of the general rules that Fick outlined in 1855.